Transformations Of Cosine Functions Analyzing G(x) = Cos(2x - Π)
In the realm of mathematics, particularly when delving into trigonometric functions, understanding transformations is paramount. This article will dissect the given function, $g(x) = cos(2x - π)$, to pinpoint the specific transformations applied to the parent cosine function. We'll explore the concepts of horizontal shifts, horizontal stretches and compressions, and phase shifts to accurately describe the changes resulting in the function $g(x)$. Our focus will be on providing a comprehensive explanation that clarifies the often-confusing aspects of trigonometric transformations, ensuring a clear understanding for anyone venturing into this domain. By meticulously analyzing the function's equation, we aim to demystify the process of identifying transformations, equipping readers with the knowledge to confidently tackle similar problems.
Decoding the Transformation
To decipher the transformations, we need to rewrite the function $g(x)$ in a standard form that reveals each transformation explicitly. The given function is:
We can factor out the coefficient of $x$ within the cosine argument:
Now, the function is in a form that clearly shows the transformations. Comparing this to the parent cosine function, $f(x) = cos(x)$, we can identify two key transformations:
- Horizontal Compression: The factor of 2 multiplying $x$ inside the cosine function indicates a horizontal compression. This means the graph is compressed horizontally by a factor of $\frac{1}{2}$. The period of the transformed function is $\frac{2π}{2} = π$, which is half the period of the parent cosine function.
- Horizontal Shift (Phase Shift): The term $\frac{π}{2}$ subtracted from $x$ indicates a horizontal shift or phase shift. Specifically, it represents a shift to the right by $\frac{π}{2}$ units. This is because the transformation is of the form $cos(B(x - C))$, where $C$ represents the horizontal shift. A positive $C$ indicates a shift to the right, and a negative $C$ indicates a shift to the left.
Therefore, the function $g(x) = cos(2(x - \frac{π}{2}))$ represents the parent cosine function compressed horizontally by a factor of $\frac{1}{2}$ and shifted horizontally to the right by $\frac{π}{2}$ units. Understanding these transformations is crucial for accurately graphing and analyzing trigonometric functions. The horizontal compression alters the period of the function, while the horizontal shift repositions the graph along the x-axis. Together, these transformations significantly change the appearance and behavior of the original cosine function.
Analyzing the Options
Now, let's analyze the given options in light of our analysis:
A. a horizontal shift right $"pi"$ units
B. a horizontal stretch by a factor of 2
C. a horizontal shift left $"pi"$ units
D. a horizontal stretch by a factor of $\frac{1}{2}$
Based on our detailed analysis, we determined that the function $g(x) = cos(2(x - \frac{π}{2}))$ undergoes a horizontal shift to the right by $"pi"/2$ units and a horizontal compression by a factor of $"1/2"$. Therefore, option A, "a horizontal shift right $"pi"$ units," is incorrect. Option B, "a horizontal stretch by a factor of 2," is also incorrect as we observed a horizontal compression, not a stretch. Option C, "a horizontal shift left $"pi"$ units," is the opposite of what we found. Option D, "a horizontal stretch by a factor of $"1/2"$," partially describes the transformation but misses the crucial horizontal shift.
To accurately select the correct answer, we must consider the combined effect of the horizontal compression and the horizontal shift. The horizontal compression by a factor of $"1/2"$ changes the period of the cosine function, and the horizontal shift by $"pi"/2$ moves the graph along the x-axis. The question asks for a phrase that describes one of the transformations. While the compression is a valid transformation, the provided options do not explicitly address the shift by $"pi"/2$. Therefore, we must re-evaluate the given options in the context of the complete transformation.
Looking at the factored form $g(x) = cos(2(x - \frac{π}{2}))$, it's clear that the horizontal shift is indeed $"pi"/2$ units to the right. However, if we consider the original form $g(x) = cos(2x - π)$, we can interpret the argument $2x - π$ differently. We can think of this as applying the transformations in a different order. First, a horizontal compression by a factor of $"1/2"$ would transform $cos(x)$ to $cos(2x)$. Then, the $-"pi"$ term could be interpreted as a horizontal shift. To determine the magnitude of this shift, we need to consider the effect of the compression. The compression affects the scale of the shift. The shift appears to be $"pi"$ units, but due to the compression by a factor of $"1/2"$, the effective shift is $"pi"/2$ units after factoring out the 2. This is because the shift is happening in the compressed domain.
The Correct Interpretation and Answer
Considering the original form $g(x) = cos(2x - π)$, let’s re-examine the horizontal shift. The argument of the cosine function is $2x - π$. To isolate the horizontal shift, we factor out the 2:
This reveals a horizontal shift of $"pi"/2$ units to the right. However, the initial interpretation of the $-"pi"$ term suggests a potential point of confusion. To clarify, let’s consider the graph of $y = cos(2x)$. This is the parent cosine function compressed horizontally by a factor of $"1/2"$. Now, we want to transform this to $y = cos(2x - π)$. The $-"pi"$ term inside the cosine function represents a phase shift. To determine the equivalent horizontal shift, we set the argument equal to zero:
Solving for $x$:
This confirms that the horizontal shift is $"pi"/2$ units to the right. Therefore, **Option A,
